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On optimal L p regularity in evolution equations

Alessandra Lunardi — 1999

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Using interpolation techniques we prove an optimal regularity theorem for the convolution u t = 0 t T t - s f s d s , where T t is a strongly continuous semigroup in general Banach space. In the case of abstract parabolic problems – that is, when T t is an analytic semigroup – it lets us recover in a unified way previous regularity results. It may be applied also to some non analytic semigroups, such as the realization of the Ornstein-Uhlenbeck semigroup in L p R n , 1 < p < , in which case it yields new optimal regularity results in fractional...

On a class of elliptic operators with unbounded coefficients in convex domains

Giuseppe Da PratoAlessandra Lunardi — 2004

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We study the realization A of the operator A = 1 2 - ( D U , D ) in L 2 Ω , μ , where Ω is a possibly unbounded convex open set in R N , U is a convex unbounded function such that lim x Ω , x Ω U x = + and lim x + , x Ω U x = + , D U x is the element with minimal norm in the subdifferential of U at x , and μ d x = c exp - 2 U x d x is a probability measure, infinitesimally invariant for A . We show that A , with domain D A = u H 2 Ω , μ : D U , D u L 2 Ω , μ is a dissipative self-adjoint operator in L 2 Ω , μ . Note that the functions in the domain of A do not satisfy any particular boundary condition. Log-Sobolev and Poincaré inequalities allow...

Computation of bifurcated branches in a free boundary problem arising in combustion theory

Olivier BaconneauClaude-Michel BraunerAlessandra Lunardi — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a parabolic Free Boundary Problem, with jump conditions at the interface. Its planar travelling-wave solutions are orbitally stable provided the bifurcation parameter u * does not exceed a critical value u * c . The latter is the limit of a decreasing sequence ( u * k ) of bifurcation points. The paper deals with the study of the bifurcated branches from the planar branch, for small . Our technique is based on the elimination of the unknown front, turning the problem into a one, to which...

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